4 ms·

Homotopy Equivalences

80 points by ibobev 1y ago

skulk 1y ago
> In fact such a 2-sphere can be wrapped around the core an arbitrary number of times. This is really hard for me to visualize. What does it look like for a 2-sphere to wrap around the core multiple times? Also, I would have expected it to be able to wrap around in multiple ways since there are more dimensions here, leading to pi^2(b^3 \ {0}) = Z^2. How would one even prove that this isn't the case?
- semolinapudding 1y ago
  There is a nice illustration of a 2-sphere wrapped twice around another 2-sphere on the Wikipedia article for the homotopy groups of spheres [0]. Now, there are many ways of proving that there is only one way (up to homotopy) of wrapping a 2-sphere n times around another 2-sphere, but all of them are fairly involved. The simplest proof comes from an analysis of the Hopf fibration, which roughly describes a relation between the 1-sphere, the 2-sphere and the 3-sphere [1]. Other than this, it follows from the theory of degrees for continuous mappings, or from the Freudenthal suspension theorem and some basic homological computations. [0] https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#/media/File:Sphere_wrapped_round_itself.png https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#/me... [1] https://en.wikipedia.org/wiki/Hopf_fibration https://en.wikipedia.org/wiki/Hopf_fibration
- pfortuny 1y ago
  This answer is probably a bit convoluted and possible erroneous. Assume the Earth has radius 2. Use coordinates (t,z) to denote “longitude” and “latitude from the North pole”. Thus (0,0) is the North Pole, (0, pi/2) is the Greenwich equatorial point and (0, pi) is the South pole. You can have “two” spheres wrapped within the Earth with the following parametrization. Using a first coordinate r to denote the distance to the Earth’s center, so that (1,t,z) denotes the points in the sphere of radius 1: (a,b)-> (1+cos(b)/2, a,b), for a,b in the interval [0,2pi]. Those are not proper spheres (the radius changes) but the surface so parametrized is homotopic to a sphere “counted two times”. It is not possible to have a warped sphere which does not cross itself, as far as I can tell (but I might be wrong). The wikipedia image linked by a sibling comment did not help me… ETA: the issue is not the dimension (2) of your spheres but the codimension (1) inside the object, and the fact that you have only removed the center of the main sphere. I think (caveat emptor) that if you remove 2 points form the solid sphere, you get Z^2. Similar to the case of surfaces and holes.
- lying4fun 1y ago
  this great visualisation of homotopy groups might be helpful A Sphere is a Loop of Loops (Visualizing Homotopy Groups) https://youtube.com/watch?v=CxGtAuJdjYI https://youtube.com/watch?v=CxGtAuJdjYI
  - bluepoint 1y ago
    How dis he do all these animations? Any idea?
    - lying4fun 1y ago
      pretty sure it’s the manim library 3b1b uses (unable to check for certain right now)
coderatlarge 1y ago
in terry tao’s recent interview with lex fridman there’s an interesting bit on poincaré conjecture where he goes out of his way not to use these words.
- randomtoast 1y ago
  It's a good (and long) interview, and I genuinely enjoyed it. Terry Tao comes across as a truly nice person. However, I noticed that he tends to be somewhat non-committal in his responses. For each question posed, he provides thorough explanations that most with a basic understanding of math can follow. Nevertheless, he rarely makes predictions or offers his opinion. He frequently ends with a remark such as, "Yes, well, it's a challenging problem." I completely understand where he is coming from. While it's true that "we don't know what we don't know", I would appreciate hearing more about his (opinionated) thoughts regarding the topics discussed during the interview.
  - williamstein 1y ago
    Fascinating observation. Maybe he is better at research partly by being disciplined to not have such opinions. Having an opinion can bias one’s approach to a problem, making it harder to solve.
    - coderatlarge 1y ago
      maybe a more mathematical interviewer could hove drawn out more predictions. i appreciate lex for having invited tao. i hope he manages to convince perelman.
      - xanderlewis 1y ago
        Just about anyone would be a more mathematical interviewer than Fridman. Even when it comes to CS, it’s blatantly obvious he doesn’t know what he’s talking about. How he got famous is such a mystery…
        1y ago
hackandthink 1y ago
"Grothendieck conjectured that the infinity groupoid captures all information about a topological space up to weak homotopy equivalence" The homotopy hypothesis has something mystical about it. https://math.ucr.edu/home/baez/homotopy/homotopy.pdf https://math.ucr.edu/home/baez/homotopy/homotopy.pdf
m_j_g 1y ago
vaguely related : synthetic homotopies visualisation tool - https://github.com/marcinjangrzybowski/cubeViz2 https://github.com/marcinjangrzybowski/cubeViz2

80 points by ibobev 1y ago

> In fact such a 2-sphere can be wrapped around the core an arbitrary number of times. This is really hard for me to visualize. What does it look like for a 2-sphere to wrap around the core multiple times? Also, I would have expected it to be able to wrap around in multiple ways since there are more dimensions here, leading to pi^2(b^3 \ {0}) = Z^2. How would one even prove that this isn't the case?

semolinapudding 1y ago
There is a nice illustration of a 2-sphere wrapped twice around another 2-sphere on the Wikipedia article for the homotopy groups of spheres [0]. Now, there are many ways of proving that there is only one way (up to homotopy) of wrapping a 2-sphere n times around another 2-sphere, but all of them are fairly involved. The simplest proof comes from an analysis of the Hopf fibration, which roughly describes a relation between the 1-sphere, the 2-sphere and the 3-sphere [1]. Other than this, it follows from the theory of degrees for continuous mappings, or from the Freudenthal suspension theorem and some basic homological computations. [0] https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#/media/File:Sphere_wrapped_round_itself.png https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#/me... [1] https://en.wikipedia.org/wiki/Hopf_fibration https://en.wikipedia.org/wiki/Hopf_fibration
pfortuny 1y ago
This answer is probably a bit convoluted and possible erroneous. Assume the Earth has radius 2. Use coordinates (t,z) to denote “longitude” and “latitude from the North pole”. Thus (0,0) is the North Pole, (0, pi/2) is the Greenwich equatorial point and (0, pi) is the South pole. You can have “two” spheres wrapped within the Earth with the following parametrization. Using a first coordinate r to denote the distance to the Earth’s center, so that (1,t,z) denotes the points in the sphere of radius 1: (a,b)-> (1+cos(b)/2, a,b), for a,b in the interval [0,2pi]. Those are not proper spheres (the radius changes) but the surface so parametrized is homotopic to a sphere “counted two times”. It is not possible to have a warped sphere which does not cross itself, as far as I can tell (but I might be wrong). The wikipedia image linked by a sibling comment did not help me… ETA: the issue is not the dimension (2) of your spheres but the codimension (1) inside the object, and the fact that you have only removed the center of the main sphere. I think (caveat emptor) that if you remove 2 points form the solid sphere, you get Z^2. Similar to the case of surfaces and holes.
lying4fun 1y ago
this great visualisation of homotopy groups might be helpful A Sphere is a Loop of Loops (Visualizing Homotopy Groups) https://youtube.com/watch?v=CxGtAuJdjYI https://youtube.com/watch?v=CxGtAuJdjYI
- bluepoint 1y ago
  How dis he do all these animations? Any idea?
  - lying4fun 1y ago
    pretty sure it’s the manim library 3b1b uses (unable to check for certain right now)

coderatlarge 1y ago

in terry tao’s recent interview with lex fridman there’s an interesting bit on poincaré conjecture where he goes out of his way not to use these words.

randomtoast 1y ago
It's a good (and long) interview, and I genuinely enjoyed it. Terry Tao comes across as a truly nice person. However, I noticed that he tends to be somewhat non-committal in his responses. For each question posed, he provides thorough explanations that most with a basic understanding of math can follow. Nevertheless, he rarely makes predictions or offers his opinion. He frequently ends with a remark such as, "Yes, well, it's a challenging problem." I completely understand where he is coming from. While it's true that "we don't know what we don't know", I would appreciate hearing more about his (opinionated) thoughts regarding the topics discussed during the interview.
- williamstein 1y ago
  Fascinating observation. Maybe he is better at research partly by being disciplined to not have such opinions. Having an opinion can bias one’s approach to a problem, making it harder to solve.
  - coderatlarge 1y ago
    maybe a more mathematical interviewer could hove drawn out more predictions. i appreciate lex for having invited tao. i hope he manages to convince perelman.
    - xanderlewis 1y ago
      Just about anyone would be a more mathematical interviewer than Fridman. Even when it comes to CS, it’s blatantly obvious he doesn’t know what he’s talking about. How he got famous is such a mystery…
      - 1y ago

hackandthink 1y ago

"Grothendieck conjectured that the infinity groupoid captures all information about a topological space up to weak homotopy equivalence" The homotopy hypothesis has something mystical about it. https://math.ucr.edu/home/baez/homotopy/homotopy.pdf https://math.ucr.edu/home/baez/homotopy/homotopy.pdf

m_j_g 1y ago

vaguely related : synthetic homotopies visualisation tool - https://github.com/marcinjangrzybowski/cubeViz2 https://github.com/marcinjangrzybowski/cubeViz2